- Optimal Inapproximability Results for MAX‐CUT and Other 2‐Variable CSPs? by Subhash Khot, Guy Kindler, Elchanan Mossel and Ryan O’Donnell
- Noise stability of functions with low influences: Invariance and optimality by Elchanan Mossel, Ryan O’Donnell and Krzysztof Oleszkiewicz
Khot and Oded Regev show that under the unique games conjecture that essentially the best algorithm for approximating vertex cover is to take all the vertices involved in a maximal matching.
Prasad Raghavendra gives a simple semidefinite programming approximation algorithm for any constraint satisfaction problem which is optimal under the UGC.
Sanjeev Arora, Boaz Barak and David Steurer describe an algorithm that given a unique game where 1-δ fraction of the edges can be satisfied, you can in time 2npoly(δ) find a coloring that satisfies a constant fraction of edges. This may or may not give evidence against the UGC.
Luca Trevisan has a nice recent survey on the unique games conjecture, covering much of the above and more, including beautiful connections between unique games and semidefinite programming.